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Social Choice and Welfare

Erschienen in:

01.09.2015 | Original Paper

Hyper-stable social welfare functions

verfasst von: Jean Lainé, Ali Ihsan Ozkes, Remzi Sanver

Erschienen in: Social Choice and Welfare | Ausgabe 1/2016

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Abstract

We define a new consistency condition for neutral social welfare functions, called hyper-stability. A social welfare function (SWF) selects a weak order from a profile of linear orders over any finite set of alternatives. Each profile induces a profile of hyper-preferences, defined as linear orders over linear orders, in accordance with the betweenness criterion: the hyper-preference of some order P ranks order Q above order Q’ if the set of alternative pairs P and Q agree on contains the one P and Q’ agree on. A special sub-class of hyper-preferences satisfying betweenness is defined by using the Kemeny distance criterion. A neutral SWF is hyper-stable (resp. Kemeny-stable) if given any profile leading to the weak order R, at least one linear extension of R is ranked first when the SWF is applied to any hyper-preference profile induced by means of the betweenness (resp. Kemeny) criterion. We show that no scoring rule is hyper-stable, unless we restrict attention to the case of three alternatives. Moreover, no unanimous scoring rule is Kemeny-stable, while the transitive closure of the majority relation is hyper-stable.

Vorheriger Artikel Cooperative decision-making for the provision of a locally undesirable facility

Nächster Artikel Statistical evaluation of voting rules

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This is what prevails in the Eurovision song contest, where ballots are based on a partial scoring method.

The Kemeny distance between two linear orders is the number of pairs of alternatives which they disagree on.

An SWF \(\alpha \) is Condorcet if at any profile, it ranks alternatives as in the majority tournament whenever the latter is a linear order.

To see why, label alternatives as x, y, and z, and consider the three weak orders \(R_{1}\), \(R_{2},\) and \(R_{3}\) (with respective asymmetric parts \( P_{1}\), \(P_{2},\) and \(P_{3}\)) defined by \(zP_{1}yP_{1}x\), \(yP_{2}zP_{2}x,\) and \(yR_{3}zP_{3}x\). Denote by \(\succsim _{1}\), \(\succsim _{2},\) and \( \succsim _{3}\) the respective hyper-preferences induced on \(\{R_{1},R_{2},\ R_{3}\}\) by \(R_{1}\), \(R_{2}\), and \(R_{3}\). Then one gets \(R_{1}\succ _{1}R_{3}\succ _{1}R_{2}\), \(R_{2}\succ _{2}R_{3}\succ _{2}R_{1},\) and \( R_{1}\thicksim _{3}R_{2}\thicksim _{3}R_{3}\). It is easily seen that for each of the 13 possible weak orders, \(R_{1}\), \(R_{2}\), and \(R_{3}\) are ranked as in \(\succsim _{1}\), \(\succsim _{2}\), or \(\succsim _{3}\). Hence, any basic profile generates a hyper-profile over the triple \(\{R_{1},R_{2},\ R_{3}\}.\)

Given a profile involving an odd number of individuals, the majority tournament is the complete and asymmetric binary relation obtained by pairwise comparisons of alternatives according to the simple majority rule. Moreover, the (necessarily unique) Condorcet winner of that profile is the alternative which defeats all other alternatives in the majority tournament.

One may also think of hyper-preferences also as preferences of individuals over others in the society.

See Igersheim (2007). The reader may refer to Jeffrey (1974), Sen (1977) and McPherson (1982) for further discussion on the more general concept of a meta-preference.

A social choice function picks one alternative at every profile of preferences over alternatives. For further studies of self-selectivity, see Koray and Unel (2003) and Koray and Slinko (2008).

An SCF maps any profile of linear orders over any finite set to an element of that set. Defining neutrality along the same lines as for SWFs allows to formally define an SCF as a function \(F:\cup _{m,n\in {\mathbb {N}}}{\mathcal {L}} (A_{m})^{n}\rightarrow \cup _{m\in {\mathbb {N}}}A_{m}\) such that for all \( n,m\in {\mathbb {N}}\) and all \(P_{N}\ \in {\mathcal {L}}(A_{m})^{n}\), \({ F(P_{N})\in A_{m}}\).

An SCF F is dictatorial if \(\exists 1\le i\le n\) such that, for all \( P_{N}\in {\mathcal {L}}(A_{m})^{n}\), \(F(P_{N})=a\Leftrightarrow aP_{i}b\) for all \(b\in A_{m}\backslash \{a\}\). Moreover, F is unanimous if for any m, for any \(P_{N}\in {\mathcal {L}}(A_{m})^{n}\), for all \(a,b\in A_{m}\), \([aP_{i}b\) for all \(1\le i\le n]\Rightarrow b\notin F(P_{N})\).

A score vector \(S^{m}\) is normalized if \(s^{1,m}=1\).

A scoring rule \(\alpha \) is convex if, for any \(m\in {\mathbb {N}}\), the score vector \(S_{\alpha }^{m}=(s_{\alpha }^{1,m},\ldots ,s_{\alpha }^{m,m})\) is such that \((s_{\alpha }^{1,m}-s_{\alpha }^{2,m})\ge (s_{\alpha }^{2,m}-s_{\alpha }^{3,m})\ge \cdots \ge (s_{\alpha }^{m-1,m}-s_{\alpha }^{m,m})\).

A social welfare correspondence is a mapping \(\delta \) from \({{\overset{}{ \underset{n,m\in {\mathbb {N}}}{\cup }}}}{\mathcal {L}}(A_{m})^{n}\) to \({{\overset{ }{\underset{m\in {\mathbb {N}}}{\cup }}}}2^{{{\mathcal {R}}(A}_{m}{)}}\backslash \varnothing \) such that, for any \(n,m\in {\mathbb {N}}\), for any \(P_{N}\ \in {\mathcal {L}}(A_{m})^{n}\), \(\delta (P_{N})\in 2^{{{\mathcal {R}}(A}_{m}{)} }\backslash \varnothing \), where \(2^{{{\mathcal {R}}(A}_{m}{)}}\backslash \varnothing \) is the set of all non-empty subsets of weak orders over \(A_{m}\).

Note that \(S(\theta , P_{N})\) is the unique finest scaling decomposition of the tournament \(\mu (P_{N})\). See Laslier (1997), Theorem 1.3.13, p. 20.

Binmore K (1975) An example in group preference. J Econ Theory 10(3):377–385CrossRef

Bossert W, Sprumont Y (2014) Strategy-proof preference aggregation. Games Econ Behav 85:109–126CrossRef

Bossert W, Storcken T (1992) Strategy-proofness of social welfare functions: the use of the Kemeny distance between preference orderings. Soc Choice Welf 9(4):345–360CrossRef

Coffman KB (2014) Representative democracy and the implementation of majority-preferred (Working paper)

Igersheim H (2007) Du paradoxe libéral-parétien à un concept de m étaclassement des préférences. Recherches é conomiques de Louvain 73(2):173–192CrossRef

Jeffrey RC (1974) Preferences among preferences. J Philos 71(13):377–391CrossRef

Koray S (2000) Self-selective social choice functions verify Arrow and Gibbard-Satterthwaite theorems. Econometrica 68(1):981–995CrossRef

Koray S, Slinko A (2008) Self-selective social choice functions. Soc Choice Welf 31(1):129–149CrossRef

Koray S, Unel B (2003) Characterization of self-selective social choice functions on the tops-only domain. Soc Choice Welf 20(3):495–507CrossRef

Laffond G, Lainé J (2000) Majority voting on orders. Theory Decis 49(3):251–289CrossRef

Lainé J (2015) Hyper-stable collective rankings. In: Mathematical social sciences (in press)

Laslier JF (1997) Tournament solutions and majority voting. Springer, Berlin, Heidelberg, New YorkCrossRef

McPherson MS (1982) Mill’s moral theory and the problem of preference change. Ethics 92(2):252–273CrossRef

Sen AK (1970) The impossibility of a Paretian liberal. J Political Econ 78(1):152–157CrossRef

Sen AK (1974) Choice, orderings and morality. In: Korner S (ed) Practical reason, papers and discussion. Blackwell, Oxford, pp 54–67

Sen AK (1977) Rational fools: a critique of behavioral foundations of economic theory. Philos Public Aff 6(4):317–344

Titel: Hyper-stable social welfare functions
verfasst von: Jean Lainé
Ali Ihsan Ozkes
Remzi Sanver
Publikationsdatum: 01.09.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2016
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0908-1

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